Problem 16 - Precalculus

Question

Use the given pair of functions to find and simplify expressions for the following functions and state the domain of each using interval notation. $$\bullet (g \circ f)(x)$$ $$\bullet (f \circ g)(x)$$ $$\bullet (f \circ f)(x)$$ $$f(x)=3 x-5, g(x)=\sqrt{x}$$

Step-by-Step Solution

Answer

\((g \circ f)(x) = \sqrt{3x-5}\), domain: \([\frac{5}{3}, \infty)\). \((f \circ g)(x) = 3\sqrt{x} - 5\), domain: \([0, \infty)\). \((f \circ f)(x) = 9x-20\), domain: \((- \infty, \infty)\).

Step 1: Understanding Composition of Functions

When composing two functions, such as \((g \circ f)(x)\), we substitute the output of one function \(f(x)\) into the other function \(g(x)\). This means that \((g \circ f)(x) = g(f(x))\). We'll follow this method for each part of the problem.

Step 2: Computing \((g \circ f)(x)\)

To find \((g \circ f)(x)\), we need to evaluate \(g(f(x))\). We know\(f(x) = 3x - 5\)and\(g(x) = \sqrt{x}\), so \(g(f(x)) = \sqrt{3x - 5}\).**Domain:** For \(g(f(x))\), the expression under the square root must be non-negative. Thus, \(3x - 5 \geq 0\), leading to \(x \geq \frac{5}{3}\). The domain in interval notation is \([\frac{5}{3}, \infty)\).

Step 3: Computing \((f \circ g)(x)\)

To find \((f \circ g)(x)\), we evaluate \(f(g(x))\). Since\(g(x) = \sqrt{x}\), we have\(f(g(x)) = f(\sqrt{x}) = 3(\sqrt{x}) - 5\).**Domain:** For \(f(g(x))\), the value inside the square root \(x\) must be non-negative. Thus, \(x \geq 0\). The domain in interval notation is \([0, \infty)\).

Step 4: Computing \((f \circ f)(x)\)

For \((f \circ f)(x)\), we substitute \(f(x)\) back into itself. This results in \(f(f(x)) = f(3x - 5) = 3(3x - 5) - 5\).Simplify this expression:\(= 9x - 15 - 5 = 9x - 20\).**Domain:** The domain of \(f(x)\) is all real numbers \((-\infty, \infty)\) as it's linear, so the domain of \((f \circ f)(x)\) is also \((-\infty, \infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when working with different mathematical functions. The domain represents all the possible input values (usually represented as "x") that the function can accept without any mathematical errors, such as division by zero or taking the square root of a negative number.
If a function is defined by a formula, identifying the domain requires analyzing potential restrictions.

• For a square root function like \(g(x) = \sqrt{x}\), the expression under the square root must be non-negative. This means \(x\) must be greater than or equal to zero.
• For a linear function like \(f(x) = 3x - 5\), there are no restrictions like division by zero or square roots of negative numbers, so it can accept all real numbers.

The domain is often expressed in interval notation, a concise way of stating which numbers are included. For example,

• For \(g(x)\), the domain is \([0,\infty)\).
• For \(f(x)\), it's \((−\infty, \infty)\).

When composing functions, the domain of the composition depends on both domains of the individual functions involved.

Square Root Function

The square root function is a fundamental mathematical concept often represented as \(g(x) = \sqrt{x}\). It only outputs non-negative numbers, as it represents a value whose square returns \(x\).
The critical characteristic of a square root function is that the value under the square root must be non-negative, which brings the domain constraint.

• The domain of \(g(x) = \sqrt{x}\) is \([0, \infty)\) because you cannot take the square root of a negative number in the real number system.
• It typically produces values greater than or equal to zero.

When combining functions or composing them, these domain restrictions always need consideration, as seen when computing \((g \circ f)(x) = \sqrt{3x - 5}\), requiring that \(3x - 5 \geq 0\), thus \(x \geq \frac{5}{3}\).

Linear Function

Linear functions are among the simplest forms of functions, with a general formula of \(f(x) = mx + b\), where \(m\) and \(b\) are constants. In our case, \(f(x) = 3x - 5\).
Linear functions graph as straight lines and do not have any restrictions on their domain.

• The domain of any linear function is all real numbers, expressed in interval notation as \((−\infty, \infty)\).
• Since linear equations are continuous, they smoothly cover all input values without interruptions or breaks.

An important application of a linear function is when it is used in function composition, as seen in computing \((f \circ f)(x)\). This involves replacing \(x\) in \(f(x)\) with \(f(x)\) itself, achieving more complex functions.